A video posted on YouTube seemingly confirms that the PlayStation 4's DualShock 4 controller will work on PS3. However, there appears to be a few exceptions.
The user, known as 'Udetto,' states that the controller will not function in a wireless state with PS3, and as such must be connected via a USB. He also added that not all titles will work properly; for example, he tried out FIFA 14 and claimed it performed poorly.
Something which isn't known is whether the Options/Share buttons becomes Start/Select, respectively. Also, it is worth pointing out that the face buttons on the DualShock are digital unlike the analogue ones found on its predecessor. Therefore games such as Metal Gear Solid HD Collection will not work as both MGS2 and 3 used them extensively. The full video can be seen here.
Note: We do not not the YouTube user so the validity of the claim should be questioned.